ECE 695Numerical Simulations

Lecture 20: Transfer Matrices and S4

Prof. Peter Bermel

March 1, 2017

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Outline

• Periodic solution strategy

– Stacked periodicity

– Transverse periodicity

• S-Matrix Simulations

• S4 RCWA solver

– Capabilities

– Live demonstration

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S-Matrices

• For S-matrix, connect incoming to outgoing fields from boundaries of region

• Mathematically,

𝑢(𝑝+1)

𝑑(0)=

𝑇 𝑢𝑢(𝑝)

𝑅 𝑢𝑑(𝑝)

𝑅 𝑑𝑢(𝑝)

𝑇 𝑑𝑑(𝑝)

𝑢(0)

𝑑(𝑝+1)

• For input from below:

– transmission from bottom to top given by 𝑇 𝑢𝑢(𝑝)

– reflection at bottom given by 𝑅 𝑑𝑢(𝑝)

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S-Matrices: Periodicity

• What happens if one or more layers are periodic?

• Then there are two types of coupling:

– Layer-to-layer (refractive)

– Mode-to-mode (diffractive)

• Can both be treated in a single framework?

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S-Matrices: Stacked Periodicity

• Recall that:

𝑠(𝑝) =𝑡 11(𝑝)

− 𝑡 12(𝑝)

𝑡 22(𝑝) −1 𝑡 21

(𝑝)𝑡 12(𝑝)

𝑡 22(𝑝) −1

−𝑡 22(𝑝) −1 𝑡 21

(𝑝)𝑡 22(𝑝) −1

• For quarter wave stack, express in terms of t-matrix elements:

𝑠(𝑝) =1 0

0 exp −𝑗𝛽𝑚,−𝑝

Δ𝑦𝑝𝑠(𝑝) exp 𝑗𝛽𝑚,+

𝑝Δ𝑦𝑝 0

0 1

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S-Matrices: Transverse Periodic Solution

• Divide into layers, uniform in z-direction

• Find Bloch states in each layer

• Calculate transfer function for field amplitudes

• Iteratively develop S-matrix

• Choose inputs from both sides

• Calculate resulting outputs (transmission and reflection) and losses (absorption A=1-T-R)

2/27/2017

Whittaker & Culshaw, Phys. Rev. B 60, 2610 (1999)Tikhodeev et al., Phys. Rev. B 66, 045102 (2002)

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S-Matrices: Periodic Solution Strategy

• Can use a Fourier series expansion in real space of the H-fields:

• In momentum space, can represent as:

• And then the electric field as:

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S-Matrices: Periodic Solution Strategy

• Eigenvalue equation becomes

• More compactly represented as:

• Where the eigenvectors f have a unique orthonormalitycondition:

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S-Matrices: Periodic Solution Strategy

• Can rephrase:

• Where H-fields are written as:

• And where E-fields are given by:

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S-Matrices: Periodic Solution Strategy

• Interface matrix in WC notation:

• Where:

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S-Matrix Construction: Recap

• We can relate the s-interface matrix to the t-interface matrix from before:

𝑠(𝑝) =𝑡 11(𝑝)

− 𝑡 12(𝑝)

𝑡 22(𝑝) −1 𝑡 21

(𝑝)𝑡 12(𝑝)

𝑡 22(𝑝) −1

−𝑡 22(𝑝) −1 𝑡 21

(𝑝)𝑡 22(𝑝) −1

• Then iteratively construct next S-matrix via:

𝑇 𝑢𝑢(𝑝)

𝑅 𝑢𝑑(𝑝)

𝑅 𝑑𝑢(𝑝)

𝑇 𝑑𝑑(𝑝)

= 𝑡 𝑢𝑢(𝑝)

1 − 𝑅 𝑢𝑑(𝑝−1)

𝑟 𝑑𝑢(𝑝) −1

𝑇 𝑢𝑢(𝑝−1)

𝑅 𝑢𝑑(𝑝−1)

+ 𝑇 𝑑𝑑(𝑝−1)

𝑟 𝑑𝑢(𝑝)

1 − 𝑅 𝑢𝑑(𝑝−1)

𝑟 𝑑𝑢(𝑝) −1

𝑇 𝑢𝑢(𝑝−1)

𝑟 𝑢𝑑(𝑝)

+ 𝑡 𝑢𝑢(𝑝)

𝑅 𝑢𝑑(𝑝−1)

1 − 𝑟 𝑑𝑢(𝑝)

𝑅 𝑢𝑑(𝑝−1) −1

𝑡 𝑑𝑑(𝑝)

𝑇 𝑑𝑑(𝑝−1)

1 − 𝑟 𝑑𝑢(𝑝)

𝑅 𝑢𝑑(𝑝−1) −1

𝑡 𝑑𝑑(𝑝)

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S-Matrices: Periodic Solution Strategy

• In WC’s notation:

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S-Matrix Simulations

• Transmission through triangular lattice converges as number of plane waves NG increases

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Whittaker & Culshaw, Phys. Rev. B 60, 2610 (1999)

Photonic Simulations with S4

Full-wave photonic simulations of arbitrary layered media, including thin-film and crystalline

PV cells

V. Liu, S. Fan, Comp. Phys. Comm. 183, 2233 (2012)

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Accuracy improves systematically with computing power

V. Liu, S. Fan, Comp. Phys. Comm. 183, 2233 (2012)

Photonic Simulations with S4

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S4: Lua Control Files

• Obtain a new, blank simulation object with no solutions:

S = S4.NewSimulation()

• Define all materials:

S:AddMaterial('name', {eps_real, eps_imag})

• Add all layers:

S:AddLayer('name', thickness, 'material_name')

• Add patterning to layers:

S:SetLayerPatternCircle('layer_name', 'inside_material', {center_x, center_y}, radius)

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S4: FMM Formulations

• Specify the excitation mechanism:

S:SetExcitationPlanewave(

{angle_phi, angle_theta}, -- phi in [0,180), theta in [0,360)

{s_pol_amp, s_pol_phase}, -- phase in degrees

{p_pol_amp, p_pol_phase})

• Specify the operating frequency:

S:SetFrequency(0.4)

• Obtain desired output:

forward_power, backward_power = S:GetPoyntingFlux('layer_name', z_offset)

print(forward_power, backward_power)

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S4: Input

Can choose several examples drawn from the literature

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S4: Output

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Transmission through multilayer stack matches analytical expression

S4: Output

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Transmission through 1D square grating of silicon and air

S4: Output

• Transmission from Fig. 4 of Tikhodeev et al., Phys. Rev. B 66, 045102 (2002).

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S-Matrix Method: Advantages

• No ad hoc assumptions regarding structures

• Applicable to wide variety of problems

• Suitable for eigenmodes or high-Q resonant modes at single frequency

• Can treat layers with large difference in length scales

• Computationally tractable enough on single core machines

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S-Matrix Method: Disadvantages

• Accurate solutions obtained more slowly as the following increase:

– Number of layers

– Absolute magnitude of Fourier components (especially for metals)

– Number of plane-wave components (~N3)

• Relatively slow for broad-band problems (time-domain is a good alternative)

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Next Class

• Is Friday, Mar. 3

• Next time, we will continue with transfer matrix models, focusing on CAMFR

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